# On integrands and loop momentum in string and field theory

## TL;DR

This paper investigates the concept of a global integrand in string theory, clarifies the definition of loop momentum, and explores its implications for loop monodromies and color-kinematics duality in gauge and gravity theories.

## Contribution

It provides a precise definition of loop momentum in string theory and constructs the associated global integrand, linking it to recent developments in loop monodromies and dualities.

## Key findings

- Defined loop momentum in string theory.
- Constructed the global integrand in the field theory limit.
- Derived monodromy relations at two and three loops.

## Abstract

The notion of a unique integrand does not a priori makes sense in field theory: different Feynman diagrams have different loop momenta and there should be no reason to compare them. In string theory, however, a global integrand is natural and allows, for instance, to make explicit the separation between left and right-moving degrees of freedom. However, the significance of this integrand had not really been investigated so far. It is even more important in view of the recently discovered loop monodromies that are related to the duality between color and kinematics in gauge and gravity loop amplitudes. This paper intends to start filling this gap, by presenting a careful definition of the loop momentum in string theory, and describing precisely the resulting global integrand obtained in the field theory limit. We will then apply this technology to write down some monodromy relations at two and three loops, and make contact with the color/kinematics duality.